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f[u-%&TITLEG6#Qcologarithmic~graph~of~eigenvalues~vs.~normal~derivativ
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45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "
#Below I'm hard-coding the points at which each grouping has it's mini
mum." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "#There is an inter
esting recursion here.  Let's say k[i] = log(-dn[r]) for some number r
.  Then k[i+2] = log(-dn[2^i + r]).  Kind of reminiscent of Fibonacci \+
recursion to me.  The same thing goes for the j's.  The j sequences wh
ich produce minimums are those with the most +'s." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "k[1]:=log(El[1]): j[1]:=log(-dn[1]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k[2]:=log(El[2]): j[2]:=log(
-dn[2]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k[3]:=log(El[3]
): j[3]:=log(-dn[3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "k[
4]:=log(El[6]): j[4]:=log(-dn[6]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "k[5]:=log(El[11]): j[5]:=log(-dn[11]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "k[6]:=log(El[22]): j[6]:=log(-dn[22
]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "k[7]:=log(El[43]): j
[7]:=log(-dn[43]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "k[8]:
=log(El[86]): j[8]:=log(-dn[86]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "k[9]:=log(El[171]): j[9]:=log(-dn[171]):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "k[10]:=log(El[342]): j[10]:=log(-dn
[342]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "#Now we make a new list with only t
he minimum points, so that we can graph only them." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 34 "list2:=[seq([k[i],j[i]],i=6..10)]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "list2;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7'7$$\"5S6ZX/82ts6!#=$\"51:UmV&R?Lu%!#>7$$\"5f2(G,:MJLL
\"F'$\"5%4&o'p,I=<D&F*7$$\"5]\"3Wyfh*Q%\\\"F'$\"5E28?-lLNjdF*7$$\"5UpS
kyc_Hb;F'$\"5/L;Ft.$4RF'F*7$$\"5lgfN-m<D;=F'$\"5ea<?Ho[#[y'F*" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "#And then we graph the minimum points." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "pointplot(list2, title=\"Minimum Po
ints Graph\", labels=[\"Eigenvalues\",\"Normal Derivatives\"]);" }}
{PARA 13 "" 1 "" {GLPLOT2D 382 382 382 {PLOTDATA 2 "6%-%'POINTSG6'7$$
\"5S6ZX/82ts6!#=$\"51:UmV&R?Lu%!#>7$$\"5f2(G,:MJLL\"F)$\"5%4&o'p,I=<D&
F,7$$\"5]\"3Wyfh*Q%\\\"F)$\"5E28?-lLNjdF,7$$\"5UpSkyc_Hb;F)$\"5/L;Ft.$
4RF'F,7$$\"5lgfN-m<D;=F)$\"5ea<?Ho[#[y'F,-%+AXESLABELSG6$Q,Eigenvalues
6\"Q3Normal~DerivativesFE-%&TITLEG6#Q5Minimum~Points~GraphFE" 1 2 0 1 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 146 "#Now we find the slope using least squares method.  \+
It's somewhat inaccurate since we're only using 7 digits and some of t
hese numbers are x*10^5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"LeastSquares(list2,v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"5R%p?J
gRd=-\"!#>\"\"\"*&$\"54;7*4rC\"*G<$!#?F'%\"vGF'F'" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "list3:=[seq([k[i],j[i]],i=5..10)]:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "pointplot(list3, title=\"Min
imum Points Graph\", labels=[\"Eigenvalues\",\"Normal Derivatives\"]);
" }}{PARA 13 "" 1 "" {GLPLOT2D 382 382 382 {PLOTDATA 2 "6%-%'POINTSG6(
7$$\"559$pg%R[v55!#=$\"5)yYBQl,8_A%!#>7$$\"5S6ZX/82ts6F)$\"51:UmV&R?Lu
%F,7$$\"5f2(G,:MJLL\"F)$\"5%4&o'p,I=<D&F,7$$\"5]\"3Wyfh*Q%\\\"F)$\"5E2
8?-lLNjdF,7$$\"5UpSkyc_Hb;F)$\"5/L;Ft.$4RF'F,7$$\"5lgfN-m<D;=F)$\"5ea<
?Ho[#[y'F,-%+AXESLABELSG6$Q,Eigenvalues6\"Q3Normal~DerivativesFJ-%&TIT
LEG6#Q5Minimum~Points~GraphFJ" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "LeastSquares(list3,v);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&$\"5*o2(4fMDl;5!#>\"\"\"*&$\"5;^*>pXKTh<$!#?F'%\"vGF'F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "list4:=[seq([k[i],j[i]],i=7.
.10)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "pointplot(list4, \+
title=\"Minimum Points Graph\", labels=[\"Eigenvalues\", \"Normal Deri
vatives\"]);" }}{PARA 13 "" 1 "" {GLPLOT2D 382 382 382 {PLOTDATA 2 "6%
-%'POINTSG6&7$$\"5f2(G,:MJLL\"!#=$\"5%4&o'p,I=<D&!#>7$$\"5]\"3Wyfh*Q%
\\\"F)$\"5E28?-lLNjdF,7$$\"5UpSkyc_Hb;F)$\"5/L;Ft.$4RF'F,7$$\"5lgfN-m<
D;=F)$\"5ea<?Ho[#[y'F,-%+AXESLABELSG6$Q,Eigenvalues6\"Q3Normal~Derivat
ivesF@-%&TITLEG6#Q5Minimum~Points~GraphF@" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "LeastSquares(list4,v);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&$\"5wQLa40k?>5!#>\"\"\"*&$\"5Bb<Ap*p#\\uJ!#?F'%\"vGF'
F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "4 0 0" 
10 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }